Diffusion effects intraparticle

In this equation the entire exterior surface of the catalyst is assumed to be uniformly accessible. Because equimolar counterdiffusion takes place for stoichiometry of the form of equation 12.4.18, there is no net molar transport normal to the surface. Hence there is no convective transport contribution to equation 12.4.21. Let us now consider two limiting conditions for steady-state operation. First, suppose that the intrinsic reaction as modified by intraparticle diffusion effects is extremely rapid. In this case PA ES will approach zero, and equation 12.4.21 indicates that the observed rate per unit mass of catalyst becomes... 

Aguwa, A.A.. Patterson, J.W., Haas. C.N., and Noll, K.E. Estimation of effective intraparticle diffusion coefficients with differential reactor columns./. Water PoMut Control Fed., 56(5) 442-448,1984. 

When intraparticle diffusion is rate limiting, the kinetic behaviour of a chemically reacting system is generally different from that which would prevail if chemical reaction were rate limiting. It is therefore extremely important to develop criteria to assess whether intraparticle diffusion effects may be neglected and thus define the conditions of experiment which would reveal true chemical kinetics rather than overall kinetics disguised by intraparticle diffusion effects. 

To assess whether a reaction is influenced by intraparticle diffusion effects, Weisz and Prater [11] developed a criterion for isothermal reactions based upon the observation that the effectiveness factor approaches unity when the generalised Thiele modulus is of the order of unity. It has been shown that the effectiveness factor for all catalyst geometries and reaction orders (except zero order) tends to unity when... 

If, however, both reactions were influenced by intraparticle diffusion effects, the rate of reaction of a particular component would be given by the product of the intrinsic reaction rate, fecg, and the effectiveness factor, Tj. Substituting eqn. (6) for the effectiveness factor gives (for a first-order isothermal reaction) the overall rate as 0tanh< >. As is often the case, the molecular weights of the diffusing reactants are similar and can be... 

No experiments with variation in particle size of the silica gel have been done to study intraparticle diffusion effects. In silica gel such diffusion would be only through the pores (analogous to the macropores of a polystyrene) since the active sites lie on the internal surface. The silica gel used by Tundo had a surface area of 500 m2/g and average pore diameter of 60 A.116). Phosphonium ion catalyst 28 gave rates of iodide displacements that decreased as the 1-bromoalkane chain length increased from C4 to Cg to C16, The selectivity of 28 was slightly less than that observed with soluble catalyst hexadecyltri-n-butylphosphonium bromide U8). Consequently the selectivity cannot be attributed to intraparticle diffusional limitations. 

When intraparticle diffusion occurs, the kinetic behaviour of the system is different from that which prevails when chemical reaction is rate determining. For conditions of diffusion control 0 will be large, and then the effectiveness factor tj( 1/ tanh 0, from equation 3.15) becomes. From equation 3.19, it is seen therefore that rj is proportional to k Ul. The chemical reaction rate on the other hand is directly proportional to k so that, from equation 3.8 at the beginning of this section, the overall reaction rate is proportional to k,n. Since the specific rate constant is directly proportional to e"E/RT, where E is the activation energy for the chemical reaction in the absence of diffusion effects, we are led to the important result that for a diffusion limited reaction the rate is proportional to e E/2RT. Hence the apparent activation energy ED, measured when reaction occurs in the diffusion controlled region, is only half the true value ... 

If intraparticle diffusion effects are important and an effectiveness factor ij is employed (as in equation 3.8) to correct the chemical kinetics observed in the absence of transport effects, then it is necessary to adopt a stepwise procedure for solution. First the pellet equations (such as 3.10) are solved in order to calculate t) for the entrance to the reactor and then the reactor equation (3.87) may be solved in finite difference form, thus providing a new value of Y at the next increment along the reactor. The whole procedure may then be repeated at successive increments along the reactor. 

To have a quantitative idea of the problem of intraparticle diffusion, effectiveness factors for the two catalysts were calculated from the observed second order rate constants (based on surface area) using the "triangle method" suggested by Saterfield (4). The effectiveness factors for Monolith and Nalcomo 474 catalysts on Synthoil liquid at 371°C (700 F) were calculated to be 0.94 and 0.216, respectively. In applying the relationship between the "Thiele Modulus," 4>> and the "effectiveness factor," n> the following simplifying assumptions were made ... 

These equations apply to intraparticle diffusion effects only. Relationships similar to eqs 152 and 153 may be... 

Figure 24 illustrates the dependence of Type III selectivity on intraparticle and interphase diffusion effects by plotting the apparent overall selectivity from eqs 159, 167 and 168 for Bim/fa = 1, against the conversion of reactant Ai. From this figure, it appears that the influence of intraparticle diffusion may reduce the overall selectivity in Type III reactions by a factor of about two. Wheeler [113] reported that this degree of reduction is independent of the intrinsic selectivity factor AA . It may therefore serve as a general rule of thumb. 

The question remains as to when the various diffusion effects really influence the conversion rate in fluid-solid reactions. Many criteria have been developed in the past for the determination of the absence of diffusion resistance. In using the many criteria no more information is required than the diffusion coefficient DA for fluid phase diffusion and for internal diffusion in a porous pellet, the heat of reaction and the physical properties of the gas and the solid or catalyst, together with an experimental value of the observed global reaction rate (R ) per unit volume or weight of solid or catalyst. For the time being the following criteria are recommended. Note that intraparticle criteria are discussed in much greater detail in Chapter 6. 

JThe catalyst particles cannot be very small. The intraparticle.diffusion effects can be significant. 

Reaction at the catalyst surface, including intraparticle diffusion effects. 

Wei, J., Intraparticle diffusion effects in complex systems of first order reactions. D, The influence of diffusion on the performance of chemical reactors. J. Catal. 1,526 (1962b). 

To ensure the absence of axial dispersion [not included in Eq. (1)], the reactor length, Az, should be at least 50 particle-diameters long, typically about 1 cm for the small particles needed to avoid intraparticle diffusion effects. The bed diameter can be about 4 or 5 mm. 

Where V is the volume of aqueous metal solution, is the amount of resin, Ig is the density of the dry ion exchange resin, R is the radius of the spherical resin, B is the adsorption capacity of resin, in g-mole-metal-ion/g-resin, t is the adsorption time, is the effective intraparticle diffusion of heavy ion metals through resins, Xf is the removal fraction of heavy metal ions by ion exchange resins. 

In industrial practice, the most convenient way of accounting for mass-transfer effects is to view the penetrable catalyst particle as a pseudo-homogeneous phase. Obstruction of mass transfer by the solid material in the particle then is reflected by an "effective" intraparticle mass-transfer or diffusion coefficient that is appropriately lower than in the contacting fluid. If this approach is taken, two fundamentally different mass-transfer situations appear Mass transfer to and from the particle across an adherent boundary layer is affected by the reaction only in that the latter sets the boundary condition at the particle. Here, mass transfer and reaction are sequential and occur in different parts of the system, and the slower of the two is the bottleneck and dictates the overall rate and its temperature dependence. Within the particle, however, mass transfer and reaction occur simultaneously and in the same volume element. Here, the reaction introduces a source-or-sink term into the basic differential material balance. If the reaction is slow, it alone controls the overall rate and its temperature dependence. If mass transfer is slow, both reaction and mass transfer affect the rate, and the apparent reaction order and activation energy are the arithmetic means of those of reaction and mass transfer. 

The presence of pores, for which the observed reaction rate is lower than the kineti-cally controlled intrinsic one, in the particles or pellets affects the reaction rate due to diffusion limitations. This intraparticle diffusion effect causes a concentration gradient within the pores. If diffusion is fast, then the concentration gradient is negligible. 

In the presence of intraparticle diffusion effects, the observed rate differs from the surface intrinsic rate which is controlled only by the reaction kinetics. Therefore, the rate must be corrected by the effectiveness factor rj. 

The reaction rate constant and activation energy are also apparent in the presence of intraparticle diffusion effects. Through Arrhenius equation, we have an apparent activation energy measured, i.e. ... 

The method has the advantage that it depends on a steady-state measurement and it is not affected by finite heat transfer. Effective intraparticle diffusivities determined in this way are commonly somewhat smaller than the values derived for the same adsorbent under simitar conditions from transient uptake rate measurements. This is because blind pores, which contribute to the flux in a transient measurement, make no contribution in a Wicke-Kallenbach system. 

---